Movable Singularity and Blowup of Semi linear Wave Equation (Microlocal analysis and asymptotic analysis)

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(1)178. Movable Singularity and Blowup of Semi linear Wave Equation By. MasafumiYOSHINO *. Abstract. In this paper we shall show the blowup of the self‐similar radially symmetric solution of a semi linear wave equation. The solution satisfies the semi linear Heun equation which is called a profile equation. We construct a singular solution in terms of elliptic function and Birkhoff normal form theory.. §1.. Introduction. Let x=(x_{1}, \ldots, x_{n}), n\geq 2 be the variable in. \mathbb{R}^{n}. and. t\in \mathbb{R} .. Consider the semi. linear wave equation with focusing nonlinearity. (ı.1) where. U_{tt}-\triangle U-U^{3}=0, U=U(x, t) , x\in \mathbb{R}^{\tau\iota}, U_{tt}=\partial^{2}U/\partial t^{2}, \triangle U=(\partial^{2}/\partial x_{1}^{2}+\cdots+\partial^{2}/\partial x_{n} ^{2})U . When we consider the blowup. of a solution, one often considers the self‐similar solution U\equiv u_{\lambda}(x, t) :=\lambda u(\lambda x, \lambda t) , \lambda>0 .. In this paper, we shall consider the self‐similar solution with radial symmetry. (1.2). U :=(T-t)^{-1}u( \frac{r}{T-t}). where T>0, r^{2}=x_{1}^{2}+. \cdot\cdot\cdot. One can easily verify that. (1.3). u. +x_{n}^{2} , and. u. is a function of a single variable. y,. u=u(y) .. satisfies the semi linear Heun equation. (1-y^{2}) \frac{d^{2}u}{dy^{2} +(\frac{n-1}{y}-4y)\frac{du}{dy}-2u+u^{3}=0.. 2010 Mathematics Subject Classification(s): Primary. 35C10 ,. Secondary. 45EI0.. Key Words: movable singularity, blowup, Birkhoff normal form. Partially supported by Grant‐in‐Aid for Scientific Research (No. 20540172), JSPS, Japan. *. Hiroshima University, Hiroshima 739‐8526, Japan..

(2) 179. MASAFUMI YOSHINO. The equation (1.3) is called a profile equation. It has four fixed regular singular points at y=0, \pm 1, \infty . By the movable singularity we mean the singularity y\neq 0, \pm 1, \infty which depends on the respective solution.. The blowup of the initial value problem of the semi linear wave equation has been. studied by many authors. In contrast with these results, our object in this paper is to study the blowup phenomenon from the viewpoint of movable singularity of the profile equation. Namely, we construct a solution of the profile equation with a movable singularity at some point. This yields the existence of blowup solution of the semi linear wave equation with singularities on the characteristic cone. The idea of proof of the main theorem is to transform the corresponding Hamiltonian. system near the blowup point to a certain normal form by the similar argument as to Birkhoff normal form theory.. The full proof of the main theorem will be published. elsewhere.. This paper is organized as follows. In §2 we show the Birkhoff‐type reduction the‐ orem. In §3 we prove the existence of singular solution of the reduced profile equation satisfying the requirements of the reduction theorem in §2. Then we state our result concerning the existence of blowup solution of semi linear wave equation. §2.. Birkhoff reduction. Set \overline{A}(y)=(1-y^{2})^{-1}(y^{-1}(n-1)-4y) and write (1.3) in. \frac{d^{2}u}{dy^{2} +\~{A} (y) \frac{du}{dy}-2 —ı -y^{2}u+ \frac{u^{3} {1-y^{2} =0.. (2.l). We shall eliminate the term containing the first derivative of unknown function. w. with. u=\alpha w. where y_{0}\neq 0, \pm 1,. \infty. by introducing a new. , where. \alpha(y)=\exp(-\frac{1}{2}\int_{y_{0} ^{y} \~{A}(s)ds). (2.2). u. :. . The resultant equation is given by. w"+A(y)w+ \frac{\alpha^{2} {1-y^{2} w^{3}=0,. (2.3) where. (2.4) By setting. (2.5). A(y)= \frac{1}{2} (-\~{A}\prime -\frac{\tilde{A}^{2} {2}-\frac{4}{1-y^{2} )= \frac{n-1}{2y^{2}(1-y^{2})^{2} (\frac{3-n}{2}+y^{2}) w=q,. w'=p,. q_{1}=y. and. B(q_{1}):= \frac{\alpha^{2} {4(1-q_{1}^{2})},.

(3) 180. MOVABLE SINGULARITY AND BLOWUP. (2.3) is written in a Hamiltonian system with the Hamiltonian function H(q_{1}). H(q_{1}):= \frac{1}{2}(p^{2}+A(q_{1})q^{2})+B(q_{1})q^{4}. (2.6) Let. z_{0}\neq 0,. \pm 1 be such that. A(z_{0})B(z_{0})\neq 0 .. Set. v= \frac{1}{2}(p^{2}+A(z_{0})q^{2}) .. p_{2} be the linear combinations of q and p such that 2v. Let q_{2} and. :=p^{2}+A(z_{0})q^{2}=q_{2}p_{2} .. Then we. B(z_{0})q^{4}=B(z_{0})(\beta q_{2}+\gamma p_{2})^{4} for some nonzero constants \beta and \gamma. W^{\tau}e can write the right‐hand side as the sum of a(v)=cq_{2}^{2}p_{2}^{2} and the remaining ones uniquely, where have. c. is a certain constant. We call a(v) the resonance part. We have. \overline{H}(q_{1}, q_{2}, p_{2}). for some. which is a polynomial of. Consider the autonomous Hamiltonian. shall transform p_{1}+H(q_{1}) to. q_{2}. and. p_{1}+H(q_{1}) .. H(q_{1})=v+a(v)+\overline{H}. p_{2}.. Let \overline{c} be a nonzero constant. We. p_{1}+t'+a(v)+\overline{c}(q_{2}^{4}+p_{2}^{4})/4 formally. Indeed, we have. Theorem 2.1. There exists a formal symplectic transformation which transforms p_{1}+H. to. p_{1}+v+a(v)+\overline{c}(q_{2}^{4}+p_{2}^{4})/4.. The proof is essentially. Birkhoff^{:}s. reduction.. Next we shall give the meaning to the normal form given by Theorem 2.1. First we introduce the homology equation.. ( qı, \tilde{p}_{1},\overline{q}_{2},\overline{p}_{2}) and y=(q_{1}, p_{1}, q_{2}, p_{2}) be the original and the transformed variables, respectively. For simplicity we sometimes write y=(y_{1}, \ldots, y_{4}) . We consider the transformation x=u(y) for some u=(u_{1}\ldots. : u_{4}) . Define Let. x=. (2.7). R:=\chi_{H^{-}}, S:=\chi_{a(v)+\overline{c}(q_{2}^{4}+p_{2}^{4})/4},. where \chi_{g} denotes the Hamiltonian vector field with Hamiltonian standard symplectic structure. Write R=r(x) \frac{\partial}{\partial x} and S=s(y) \frac{\partial}{\partial y}. Define. \Lambda(y)=(1, 0, q_{2}/2, -p_{2}/2) .. Lemma 2.2. Suppose that. (2.8). u. with respect to a. Thcn wc have. satisfy the homology equation. \Lambda(y)\nabla u+s(y)\nabla u=r(u)+\Lambda(u) .. Then, the transformation x=u(y) maps the vector field. s(y) \frac{\partial}{\partial y}.. g. ( \Lambda(x)+r(x) \frac{\partial}{\partial x}. to (\Lambda(y)+. Proof.. (2.9). ( \Lambda(x)+r(x) \frac{\partial}{\partial x}=(\Lambda(u)+r(u) (\nabla u)^{-1} \frac{\partial}{\partial y}=(\Lambda(y)+s(y) \frac{\partial}{\partial y}. \square.

(4) 181 181. MASAFUMI YOSHINO. We shall solve (2.8). Define. (2.10). q_{2}=\alpha\zeta, p_{2}=\eta\zeta_{:}. where \zeta is a complex parameter. Assume that. \alpha. and. \eta. satisfy. 2c\alpha^{2}\eta^{2}(\eta-\alpha)+\overline{c}(\eta^{5}-\alpha^{5})\neq 0, \eta +\alpha\neq 0.. (2.11). Let \eta_{0}>0 be given. Set \rho=q_{2}+p_{2} and define. (2.12). \Omega_{0} :=\{(\rho, q{\imath}) | q_{1}-z_{0}|<\eta_{0}, |\rho|<\eta_{0}\}.. Then we have. Theorem 2.3. Suppose that \alpha and \eta satisfy (2.11). Then there exist6 an \eta_{0}>0 such that if p_{1} is in some neighborhood of the origin and (q_{1}, q_{2_{・}}.p_{2}) is given by (2.10) with q_{2} and p_{2} replaced by q_{2}^{-1} and p_{2}^{-1} , respectively, and \zeta=(\alpha+\eta)^{-1}\rho, (q_{1}, \rho)\in\Omega_{0}, then the vector field ( \Lambda(x)+r(x) \frac{\partial}{\partial x} is transformed to ( \Lambda(y)+s(y) \frac{\partial}{\partial y} by an analytic change of coordinates.. §3.. Movable singularity and blowup solution. In this section we shall construct a solution of (1.3) with movable singularity and state our main result. In view of Theorem 2.3 we consider the Hamiltonian p_{1}+q_{2}p_{2}+. cq_{2}^{2}p_{2}^{2}+\overline{c}(q_{2}^{4}+p_{2}^{4}) , where c\neq 0 and \overline{c}\neq 0 are constants. By setting. q_{2}=q. and. p_{2}=p. we consider the Hamiltonian. \overline{H} :=qp+\frac{\varepsilon}{2}q^{2}p^{2}-\frac{\eta}{8}(q^{2}-p^{2})^ {2},. (3.1). where \epsilon and \eta\neq 0 are constants. Because \overline{c} can be chosen arbitrarily, we may assume \epsilon\neq 0, \epsilon+\eta\neq 0 without loss of generality. Suppose that (q, p) is the solution of the Hamiltonian system for \overline{H} . Then there exists a constant C_{2} such that Define. \zeta=\frac{q+p}{2}, \xi=\frac{q-p}{2_{i} .. (3.2) Then we have. (3.3). C_{2}= \overline{H}=(\zeta^{2}+\xi^{2})+\frac{\epsilon}{2}(\zeta^{2}+\xi^{2}) ^{2}+2\eta\zeta^{2}\xi^{2}. = \frac{\epsilon+\eta}{2}(\zeta^{2}+\xi^{2}+\frac{1}{\epsilon+\eta})^{2}- \frac{1}{2(\epsilon+\eta)}-\frac{\eta}{2}(\zeta^{2}-\xi^{2})^{2}. \overline{H}(q, p)\equiv C_{2}..

(5) 182. MOVABLE SINGULARITY AND BLOWUP. Hence we have. 1= \frac{(\epsilon+\eta)^{2} {A}(\zeta^{2}+\xi^{2}+\frac{1}{\epsilon+\eta})^{2} -\frac{\eta(\epsilon+\eta)}{A}(\zeta^{2}-\xi^{2})^{2}. (3.4) where. A=1+2C_{2}(\epsilon+\eta) .. (3.5). We determine. \theta=\theta(z). such that. \sin^{2}\theta=\frac{(\epsilon+\eta)^{2} {A}(\zeta^{2}+\xi^{2}+\frac{1} {\epsilon+\eta})^{2}, \cos^{2}\theta=-\frac{\eta(\epsilon+\eta)}{A}(\zeta^{2}- \xi^{2})^{2}. Then, by (3.5) and simple computations we have. (3.6). \zeta\equiv\zeta(z)=\sqrt{\frac{\sqrt{\frac{A(\epsilon+\eta)}{-\eta} \cos\theta +\sqrt{sınA} {2(\ep\thetasilon+\eta)1} ,. (3.7). \xi\equiv\xi(z). Set. X(z)=\sin?(z)+\eta\epsilon^{-1}/\sqrt{A}. and define. \mathcal{A}=\sqrt{\mathcal{E}+\frac{i}{2}\sqrt{\mathcal{F} , \mathcal{B}=\sqrt {\mathcal{E}-\frac{i}{2}\sqrt{\mathcal{F} ,. (3.8) where. \mathcal{E}=\frac{1}{2(\epsilon+\eta)}(\sqrt{A}X(z)-\eta\epsilon^{-1}-1) \backslash \mathcal{F}=\frac{A}{\eta(\epsilon+\eta)}(1-(X(z)-\eta\epsilon^{-1}/\sqrt{A})^ {2}) .. (3.9) (3.10) Then wc. \sec. that \zeta=\mathcal{A} and \xi=\mathcal{B} . Therefore, by (3.2) we obtain. (3.ıl). q(z)=\mathcal{A}+i\mathcal{B},. p(z)=\mathcal{A}-i\mathcal{B}.. Then we have. Lemma 3.1.. X(z) is an elliptic function.. In view of (3.11) we see that the solution has movable singularity given by the elliptic function. By virtue of the parametrization via the eıliptic function, we shall construct. the blowup solution of (1.1) with singularities on some characteristic cone. Let a(v) be as in Theorem 2.1. Then we have. Theorem 3.2. Let. T>0 .. Assume that z_{0}\neq 0,. \pm 1. . Given a neighborhood \Omega_{0} of. Then there exist z_{1}\in\Omega_{0} and a blowup solution U of (1.1) such that z_{1}(T-t)=r, r^{2}=x_{1}^{2}+\cdots+x_{n}^{2} , (t_{:}x)\in \mathbb{R}^{n+1}. set. U. z_{0}.. blows up on the.

(6) 183. MASAFUMI YOSHINO. Proof. We shall look for u in (1.2) such that u satisfies (1.3) and has a movable singularity. Indeed, we inake the reduction as in Theorem 2.3 to (1.3) and we obtain the autonomous system. Indeed, the assumption (2.11) is the condition for q_{2} and p_{2}, which can be satisfied in view of the parametrization of singular solution in the above by slight change of parameters.. Therefore we obtain a singular solution of (1.3) parametrized by the solution of the autonomous equation.. Next one can easily show that there is a local diffeomorphic. change of variables in some neighborhood of. z_{0}. between the original variable \tilde{q}_{1} and. . By expressing the singular solution in terms of the variable of (1.3) we obtain the singular solution of (1.3). The location of singularity is clear in view of the definition q_{1}. of a radially symmetric self‐similar solution.. \square. References. [1] Bitzori P., Maison D. and Wasserman A., Self‐similar solutions of semi linear wave equa‐ tions with a focusing nonlinearity, Nonlinearity, 20, (2007), 2061‐2074. [2] Hurwitz, A., Vorlesungen ueber Allgemeine Funktionen‐theorie und Elliptische Funktio‐ nen, Springer‐ Verlag, Berlin Heidelberg. 1964.. [3] Ito, H., Integrable symplectic maps and their Birkhoff normal forms_{i} Tohoku Math. J. 49 (1997) ; 73 ‐114. [4] Kycia, R. A. and Filipuk, G., On the generalized Emden‐Fowler and isothermal spheres equations, Applied Mathematics and Computation, 265, (2015) 1003‐1010..

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